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Bisector of Angle between Two Vectors

A vector of m...

Question

A vector of magnitude $2$ along a bisector of the angle between the two vectors $2 \to i - 2 \to j + \to k$ and $\to i + 2 \to j - 2 \to k$ is

A

\frac{2}{\sqrt{10}} (3 \to i - \to k)

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B

\frac{1}{\sqrt{26}} (\to i - 4 \to j + 3 \to k)

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C

\frac{2}{\sqrt{26}} (\to i - 4 \to j + 3 \to k)

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D

None of these

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Solution

The correct options are
C $\frac{2}{\sqrt{26}} (\to i - 4 \to j + 3 \to k)$
D $\frac{2}{\sqrt{10}} (3 \to i - \to k)$
The vector along the angle bisector is obtained by the sum and difference of unit vectors along both arms.

Thus, $\frac{1}{3} (2 \to i - 2 \to j + \to k \pm \to i + 2 \to j - 2 \to k) = \to i - \frac{1}{3} \to k$ and $\frac{1}{3} \to i - \frac{4}{3} \to j + \to k$

Now for the vector of magnitude $2$ , we multiply the unit vector in that direction by $2$ and get $\frac{2}{\sqrt{10}} (3 \to i - \to k)$ and $\frac{2}{\sqrt{26}} (\to i - 4 \to j + 3 \to k)$

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